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Theorem pmtrdifellem3 17721
 Description: Lemma 3 for pmtrdifel 17723. (Contributed by AV, 15-Jan-2019.)
Hypotheses
Ref Expression
pmtrdifel.t 𝑇 = ran (pmTrsp‘(𝑁 ∖ {𝐾}))
pmtrdifel.r 𝑅 = ran (pmTrsp‘𝑁)
pmtrdifel.0 𝑆 = ((pmTrsp‘𝑁)‘dom (𝑄 ∖ I ))
Assertion
Ref Expression
pmtrdifellem3 (𝑄𝑇 → ∀𝑥 ∈ (𝑁 ∖ {𝐾})(𝑄𝑥) = (𝑆𝑥))
Distinct variable groups:   𝑥,𝑄   𝑥,𝑇
Allowed substitution hints:   𝑅(𝑥)   𝑆(𝑥)   𝐾(𝑥)   𝑁(𝑥)

Proof of Theorem pmtrdifellem3
StepHypRef Expression
1 pmtrdifel.t . . . . . . 7 𝑇 = ran (pmTrsp‘(𝑁 ∖ {𝐾}))
2 pmtrdifel.r . . . . . . 7 𝑅 = ran (pmTrsp‘𝑁)
3 pmtrdifel.0 . . . . . . 7 𝑆 = ((pmTrsp‘𝑁)‘dom (𝑄 ∖ I ))
41, 2, 3pmtrdifellem2 17720 . . . . . 6 (𝑄𝑇 → dom (𝑆 ∖ I ) = dom (𝑄 ∖ I ))
54adantr 480 . . . . 5 ((𝑄𝑇𝑥 ∈ (𝑁 ∖ {𝐾})) → dom (𝑆 ∖ I ) = dom (𝑄 ∖ I ))
65eleq2d 2673 . . . 4 ((𝑄𝑇𝑥 ∈ (𝑁 ∖ {𝐾})) → (𝑥 ∈ dom (𝑆 ∖ I ) ↔ 𝑥 ∈ dom (𝑄 ∖ I )))
74difeq1d 3689 . . . . . 6 (𝑄𝑇 → (dom (𝑆 ∖ I ) ∖ {𝑥}) = (dom (𝑄 ∖ I ) ∖ {𝑥}))
87unieqd 4382 . . . . 5 (𝑄𝑇 (dom (𝑆 ∖ I ) ∖ {𝑥}) = (dom (𝑄 ∖ I ) ∖ {𝑥}))
98adantr 480 . . . 4 ((𝑄𝑇𝑥 ∈ (𝑁 ∖ {𝐾})) → (dom (𝑆 ∖ I ) ∖ {𝑥}) = (dom (𝑄 ∖ I ) ∖ {𝑥}))
106, 9ifbieq1d 4059 . . 3 ((𝑄𝑇𝑥 ∈ (𝑁 ∖ {𝐾})) → if(𝑥 ∈ dom (𝑆 ∖ I ), (dom (𝑆 ∖ I ) ∖ {𝑥}), 𝑥) = if(𝑥 ∈ dom (𝑄 ∖ I ), (dom (𝑄 ∖ I ) ∖ {𝑥}), 𝑥))
111, 2, 3pmtrdifellem1 17719 . . . 4 (𝑄𝑇𝑆𝑅)
12 eldifi 3694 . . . 4 (𝑥 ∈ (𝑁 ∖ {𝐾}) → 𝑥𝑁)
13 eqid 2610 . . . . 5 (pmTrsp‘𝑁) = (pmTrsp‘𝑁)
14 eqid 2610 . . . . 5 dom (𝑆 ∖ I ) = dom (𝑆 ∖ I )
1513, 2, 14pmtrffv 17702 . . . 4 ((𝑆𝑅𝑥𝑁) → (𝑆𝑥) = if(𝑥 ∈ dom (𝑆 ∖ I ), (dom (𝑆 ∖ I ) ∖ {𝑥}), 𝑥))
1611, 12, 15syl2an 493 . . 3 ((𝑄𝑇𝑥 ∈ (𝑁 ∖ {𝐾})) → (𝑆𝑥) = if(𝑥 ∈ dom (𝑆 ∖ I ), (dom (𝑆 ∖ I ) ∖ {𝑥}), 𝑥))
17 eqid 2610 . . . 4 (pmTrsp‘(𝑁 ∖ {𝐾})) = (pmTrsp‘(𝑁 ∖ {𝐾}))
18 eqid 2610 . . . 4 dom (𝑄 ∖ I ) = dom (𝑄 ∖ I )
1917, 1, 18pmtrffv 17702 . . 3 ((𝑄𝑇𝑥 ∈ (𝑁 ∖ {𝐾})) → (𝑄𝑥) = if(𝑥 ∈ dom (𝑄 ∖ I ), (dom (𝑄 ∖ I ) ∖ {𝑥}), 𝑥))
2010, 16, 193eqtr4rd 2655 . 2 ((𝑄𝑇𝑥 ∈ (𝑁 ∖ {𝐾})) → (𝑄𝑥) = (𝑆𝑥))
2120ralrimiva 2949 1 (𝑄𝑇 → ∀𝑥 ∈ (𝑁 ∖ {𝐾})(𝑄𝑥) = (𝑆𝑥))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896   ∖ cdif 3537  ifcif 4036  {csn 4125  ∪ cuni 4372   I cid 4948  dom cdm 5038  ran crn 5039  ‘cfv 5804  pmTrspcpmtr 17684 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-om 6958  df-1o 7447  df-2o 7448  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-pmtr 17685 This theorem is referenced by:  pmtrdifel  17723  pmtrdifwrdellem3  17726
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